3 V Change of Basis 3 V 1

3. V. Change of Basis 3. V. 1. Changing Representations of Vectors 3. V. 2. Changing Map Representations

3. V. 1. Changing Representations of Vectors Definition 1. 1: Change of Basis Matrix The change of basis matrix for bases B, D V is the representation of the identity map id : V → V w. r. t. those bases. Lemma 1. 2: Proof: Alternatively, Changing Basis

Example 1. 3: →

Lemma 1. 4: A matrix changes bases iff it is nonsingular. Proof : Bases changing matrix must be invertible, hence nonsingular. Proof : (See Hefferon, p. 239. ) Nonsingular matrix is row equivalent to I. Hence, it equals to the product of elementary matrices, which can be shown to represent change of bases. Corollary 1. 5: A matrix is nonsingular it represents the identity map w. r. t. some pair of bases.

Exercises 3. V. 1. Find the change of basis matrix for B, D R 2. 1. (a) (c ) 2. B = E 2 , D = e 2 , e 1 D = E 2 (b) , B = E 2 (d ) Let p be a polynomial in P 3 with where B = 1+x, 1 x, x 2+x 3, x 2 x 3 . Find a basis D such that

3. V. 2. Changing Map Representations

Example 2. 1: Let Rotation by π/6 in x-y plane t : R 2 → R 2

Let →

Example 2. 2: → ∴ Let Then

Consider t : V → V with matrix representation T w. r. t. some basis. If basis B s. t. T = t. B → B is diagonal, Then t and T are said to be diagonalizable. Definition 2. 3: Matrix Equivalent Same-sized matrices H and H are matrix equivalent if nonsingular matrices P and Q s. t. H = P H Q or H = P 1 H Q 1 Corollary 2. 4: Matrix equivalent matrices represent the same map, w. r. t. appropriate pair of bases. Matrix equivalence classes.

Elementary row operations can be represented by left-multiplication (H = P H ). Elementary column operations can be represented by right-multiplication ( H = H Q ). Matrix equivalent operations cantain both (H = P H Q ). ∴ row equivalent matrix equivalent Example 2. 5: and are matrix equivalent but not row equivalent. Theorem 2. 6: Block Partial-Identity Form Any m n matrix of rank k is matrix equivalent to the m n matrix that is all zeros except that the first k diagonal entries are ones. Proof: Gauss-Jordan reduction plus column reduction.

Example 2. 7: G-J row reduction: Column swapping: Combined:

Corollary 2. 8: Matrix Equivalent and Rank Two same-sized matrices are matrix equivalent iff they have the same rank. That is, the matrix equivalence classes are characterized by rank. Proof. Two same-sized matrices with the same rank are equivalent to the same block partial-identity matrix. Example 2. 9: The 2 2 matrices have only three possible ranks: 0, 1, or 2. Thus there are 3 matrix-equivalence classes.

If a linear map f : V n → W m is rank k, then some bases B → D s. t. f acts like a projection Rn → Rm.

Exercises 3. V. 2. 1. Show that, where A is a nonsingular square matrix, if P and Q are nonsingular square matrices such that PAQ = I then QP = A 1. 2. Are matrix equivalence classes closed under scalar multiplication? Addition? 3. (a) If two matrices are matrix-equivalent and invertible, must their inverses be matrix-equivalent? (b) If two matrices have matrix-equivalent inverses, must the two be matrixequivalent? (c) If two matrices are square and matrix-equivalent, must their squares be matrix-equivalent? (d) If two matrices are square and have matrix-equivalent squares, must they be matrix-equivalent?